Nomination and definition

Nomination (that is, naming, giving a name to a thing) is important but underestimated stage in development of a mathematical concept. I quote Semen Kutateladze:

Nomination is a principal ingredient of education and transfer of knowledge. Nomination differs from definition. The latter implies the description of something new with the already available notions. Nomination is the calling of something, which is the starting point of any definition. Of course, the frontiers between nomination and definition are misty and indefinite rather than rigid and clear-cut.

And here is another mathematician talking about this important, but underrated concept:

Suppose that you want to teach the ‘cat’ concept to a very young child. Do you explain that a cat is a relatively small, primarily carnivorous mammal with retractible claws, a distinctive sonic output, etc.? I’ll bet not. You probably show the kid a lot of different cats, saying ‘kitty’ each time, until it gets the idea. (R. P. Boas, Can we make mathematics intelligible?, American Mathematical Monthly 88 (1981) 727-731.)

And back to Kutateladze:

We are rarely aware of the fact the secondary school arithmetic and geometry are the finest gems of the intellectual legacy of our forefathers. There is no literate who fails to recognize a triangle. However, just a few know an appropriate formal definition. This is not by chance at all, since the definition of triangle is absent in the Elements.

The list can be continued – most basic concepts of elementary mathematics is the result of nomination not supported by a formal definition: number, set, curve, figure, etc. Basically, mathematics starts with nomination. I had already have a chance to write about Vladimir Radzivilovsky and his method of teaching mathematics to very young children. It involved a systematic use of an idea — borrowed from 19th century Italian educator Maria Montessori — of teaching children to recognise and name basic geometric shapes: triangle, square, circle, etc. and comparing them by placing shapes into similarly shaped (but perhaps differently oriented) pockets in a board. Nomenclature is a key component of the Montessori Method.

Montessori with a child playing with geometric shapes.

It is important to emphasise that not only vision, but also locomotor and tactile sensory systems were engaged in this exercise –Radzivilovsky trained his children to recognise and name shapes with eyes closed.

Montessori’s inset board

Perhaps, I would suggest to introduce a name for an even more elementary didactic act: pointing, like pointing a finger at a thing before naming it. Julia Rempel’s introduction of proportions via tasting mixtures of fruit juices, described in my previous post Proportions by mouth, is an example of pointing — it starts by demonstrating physical manifestations of a mathematical concept.

I believe a clear distinction needs to be drawn between giving a name to a thing (such as a triangle or a cat) and giving a name to a process (such as the process of creating a set from a collection of things). Moreover, pedagogically, there is a difference between giving a name to a thing-in-the-world (such as a cat, or even a drawing of a cat) and giving a name to an abstract concept (such as “catness” or a set). I am sure such distinctions mean that some naming activities are easier for children than are others — but then I am a computer scientist, and we tend to concern ourselves with abstract concepts and with processes.

But Hilbert, very famously, did not DEFINE points and lines, John, in his axiomatic treatment of geometry. He took these entities as undefined primitives, and assumed (assumed!) that they satisfy his axioms. He once told a student in a bar that his theory of geometry could just as well apply to the beer mugs and tables in front of them as to anyone’s traditional conceptions of points and lines, if beer mugs and tables happened to satisfy the axioms. Whatever Hilbert was doing, it was not defining things.

I do not agree with your assessment of Hilbert’s axiomatisation: it was him who defined points and lines by listing relations between them. Euclid did not definf lines and points (in our modern understanding of “definition”). He described them.